<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd">
<html>
<head>
<title></title>
<!--Generated on Tue Jun  3 20:02:58 2014 by LaTeXML (version 0.7.999_04) http://dlmf.nist.gov/LaTeXML/.-->

<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<link rel="stylesheet" href="../../../../../../../LaTeXML.css" type="text/css">
<link rel="stylesheet" href="../../../../../../../ltx-article.css" type="text/css">
<link rel="stylesheet" href="../../../../../../../customRules.css" type="text/css">
</head>
<body>
<div class="ltx_page_main">
<div class="ltx_page_content">
<div class="ltx_document">
<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Calibration of the imaging system (astigmatism method)</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">3D SMLM imaging can be performed by introducing a weak cylindrical
lens into the imaging path to create slight astigmatism in the image
<cite class="ltx_cite">[<a href="#bib.bib14" title="Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy" class="ltx_ref">1</a>]</cite>. This results in images of molecules with different
ellipticity depending on their axial position. When a molecule is
in focus, its image appears round. If the molecule is slightly above
or below the focal plane, its image appears ellipsoidal. Calibration
of the imaging system is needed to determine the orientation of the
imaged ellipsoid (the camera chip might not be aligned with cylindrical
lens) and the relationships between the axial position and ellipticity
of the imaged molecules.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Calibration of the imaging system</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<p class="ltx_p">Calibration is a procedure which determines the orientation angle
<img id="Sx1.SSx1.p1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi12.png" width="15" height="19" alt="\phi"> of the imaged ellipsoids, and the relationship between the
axial position <img id="Sx1.SSx1.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi19.png" width="13" height="12" alt="z"> of the molecules and their imaged widths <img id="Sx1.SSx1.p1.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi14.png" width="48" height="15" alt="\sigma_{1},\sigma_{2}">.
We modeled this relationship by a pair of third degree polynomials</p>
<table id="Sx1.EGx1" class="ltx_equationgroup ltx_eqn_eqnarray">

<tr id="Sx1.E1" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi3.png" width="48" height="21" alt="\displaystyle\hat{\sigma}_{1}\left(z\right)"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.E1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.E1.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi7.png" width="245" height="25" alt="\displaystyle a_{1}\left(z-c_{1}\right)^{2}+d_{1}\left(z-c_{1}\right)^{3}+b_{1%
}\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
</tr>
<tr id="Sx1.E2" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.E2.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi5.png" width="48" height="21" alt="\displaystyle\hat{\sigma}_{2}\left(z\right)"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.E2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.E2.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi8.png" width="245" height="25" alt="\displaystyle a_{2}\left(z-c_{2}\right)^{2}+d_{2}\left(z-c_{2}\right)^{3}+b_{2%
}\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(2)</span></td>
</tr>
</table>
</div>
<div id="Sx1.SSx1.p2" class="ltx_para">
<p class="ltx_p">The calibration is typically performed using a Z-stack of images of
sub-diffraction fluorescent beads. We use a sparse sample with about
10 to 50 beads in the image and a Z-stack image sequence with an axial
range of about <img id="Sx1.SSx1.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi10.png" width="41" height="19" alt="2\,\mathrm{\mu m}"> and a step size of <img id="Sx1.SSx1.p2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi9.png" width="49" height="16" alt="10\,\mathrm{nm}">.</p>
</div>
</div>
<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Determining the orientation angle</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<ol id="I1" class="ltx_enumerate">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">1.</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p">A sequence of images from a Z-stack is processed slice-by-slice using
the methods for raw data analysis (<a href="../../filters/ui/Filters.html" title="" class="ltx_ref">image filtering</a>,
<a href="../../detectors/Detectors.html" title="" class="ltx_ref">approximate localization</a>,
<a href="Fitting.html" title="" class="ltx_ref">PSF fitting</a>). Images of the beads are fit independently
using the <a href="EllipticGaussianEstimatorUI.html" title="" class="ltx_ref">elliptical Gaussian PSF</a>
with <span class="ltx_text ltx_markedasmath"><img id="I1.i1.p1.m1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi12.png" width="15" height="19" alt="\phi"></span> as a free parameter.</p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">2.</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p">Results close to circular are discarded as the angle <img id="I1.i2.p1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi12.png" width="15" height="19" alt="\phi"> cannot
be determined.</p>
</div>
</li>
<li id="I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">3.</span> 
<div id="I1.i3.p1" class="ltx_para">
<p class="ltx_p">The final orientation angle is calculated as the circular mean of
all remaining measurements</p>
<table id="Sx1.E3" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E3.m1" class="ltx_Math" style="vertical-align:-25px" src="mi/mi6.png" width="327" height="57" alt="\phi=\frac{1}{4}\atantwo\left(\frac{1}{n}\sum_{i=1}^{n}{\sin\varphi_{i}},\,%
\frac{1}{n}\sum_{i=1}^{n}{\cos\varphi_{i}}\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(3)</span></td>
</tr>
</table>
<p class="ltx_p">where <img id="I1.i3.p1.m1" class="ltx_Math" style="vertical-align:-8px" src="mi/mi15.png" width="145" height="25" alt="\varphi_{i}=4\left(\phi_{i}\bmod\frac{\pi}{2}\right)"> adjusts
the fitted angles <img id="I1.i3.p1.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi13.png" width="21" height="19" alt="\phi_{i}">, and <img id="I1.i3.p1.m3" class="ltx_Math" style="vertical-align:-2px" src="mi/mi18.png" width="15" height="12" alt="n"> is the number of measured
beads.</p>
</div>
</li>
</ol>
</div>
</div>
<div id="Sx1.SSx3" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Ellipticity as a function of an axial position</h2>

<div id="Sx1.SSx3.p1" class="ltx_para">
<ol id="I2" class="ltx_enumerate">
<li id="I2.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">1.</span> 
<div id="I2.i1.p1" class="ltx_para">
<p class="ltx_p">Using the approximate positions of the beads in multiple Z-planes
and the orientation angle <img id="I2.i1.p1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi12.png" width="15" height="19" alt="\phi">, both determined in the previous
step, the images of the beads are fit again using the <a href="EllipticGaussianEstimatorUI.html" title="" class="ltx_ref">elliptical Gaussian PSF</a>,
but with a fixed angle <img id="I2.i1.p1.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi12.png" width="15" height="19" alt="\phi">.</p>
</div>
</li>
<li id="I2.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">2.</span> 
<div id="I2.i2.p1" class="ltx_para">
<p class="ltx_p">To estimate the coefficients <img id="I2.i2.p1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi16.png" width="93" height="19" alt="a_{1},b_{1},c_{1},d_{1}"> and <img id="I2.i2.p1.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi17.png" width="93" height="19" alt="a_{2},b_{2},c_{2},d_{2}">
in Equations (<a href="#Sx1.E1" title="(1) ‣ Calibration of the imaging system ‣ Calibration of the imaging system (astigmatism method)" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>) and (<a href="#Sx1.E2" title="(2) ‣ Calibration of the imaging system ‣ Calibration of the imaging system (astigmatism method)" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>),
we first fit the pair of polynomials for each bead separately using
an iterative least-squares algorithm which automatically discards
outliers.</p>
</div>
</li>
<li id="I2.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">3.</span> 
<div id="I2.i3.p1" class="ltx_para">
<p class="ltx_p">From the fitted models, we determine a common focal plane of the beads
as <img id="I2.i3.p1.m1" class="ltx_Math" style="vertical-align:-8px" src="mi/mi11.png" width="45" height="25" alt="\frac{c_{1}+c_{2}}{2}"> and shift the data along the <img id="I2.i3.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi19.png" width="13" height="12" alt="z">-axis
such that all beads are positioned at the same focal plane.</p>
</div>
</li>
<li id="I2.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">4.</span> 
<div id="I2.i4.p1" class="ltx_para">
<p class="ltx_p">The final coefficients are obtained by fitting the pair of polynomials
to all shifted data points. The ‘‘zero’’ axial position is given
by the intersection of the two polynomials.</p>
</div>
</li>
</ol>
</div>
</div>
<div id="Sx1.SSx4" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Guidelines for the choice of parameters</h2>

<div id="Sx1.SSx4.p1" class="ltx_para">
<p class="ltx_p">As the signal to noise ratio is usually higher in the 3D calibration
data with fluorescent beads, users should set the threshold, in the
case of the wavelet filter, to 5 to 8 times the standard deviation
of the 1st wavelet level, e.g., <span class="ltx_text ltx_font_typewriter">6*std(Wave.F1)</span>. The rest
of the settings are the same as in 2D data analysis. Use the Preview
button to see detections of the calibration beads with the current
settings.</p>
</div>
</div>
<div id="Sx1.SSx5" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx5.p1" class="ltx_para">
<ul id="I3" class="ltx_itemize">
<li id="I3.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I3.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="PSF.html" title="" class="ltx_ref">Point-spread function (PSF)</a></p>
</div>
</li>
<li id="I3.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I3.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="EllipticGaussianEstimatorUI.html" title="" class="ltx_ref">Rotated elliptical Gaussian function PSF model (3D using astigmatism)</a></p>
</div>
</li>
<li id="I3.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I3.i3.p1" class="ltx_para">
<p class="ltx_p"><a href="Fitting.html" title="" class="ltx_ref">Fitting point-spread function models</a></p>
</div>
</li>
<li id="I3.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I3.i4.p1" class="ltx_para">
<p class="ltx_p"><a href="FittingRegion.html" title="" class="ltx_ref">Definition of the fitting region</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib14" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">B. Huang, W. Wang, M. Bates and X. Zhuang</span><span class="ltx_text ltx_bib_year">(2008)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Science</span> <span class="ltx_text ltx_bib_volume">319</span> (<span class="ltx_text ltx_bib_number">5864</span>), <span class="ltx_text ltx_bib_pages"> pp. 810–3</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1126/science.1153529" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Calibration of the imaging system (astigmatism method)" class="ltx_ref"><span class="ltx_text ltx_ref_title">Calibration of the imaging system (astigmatism method)</span></a>.
</span>
</li>
</ul>
</div>
</div>
</div>
<div class="ltx_page_footer">
<div class="ltx_page_logo">Generated  on Tue Jun  3 20:02:58 2014 by <a href="http://dlmf.nist.gov/LaTeXML/">LaTeXML <img src="" alt="[LOGO]"></a>
</div>
</div>
</div>
</body>
</html>
